A new version of transfer matrix method for multibody systems

In order to avoid the global dynamics equations and increase the computational efficiency for multibody system dynamics (MSD), the transfer matrix method of multibody system (MSTMM) has been developed and applied very widely in research and engineering in recent 20 years. It differs from ordinary methods in multibody system dynamics with respect to the feature that there is no need for a global dynamics equation, and it uses low-order matrices for high computational efficiency. For linear systems, MSTMM is exact even if continuous elements like beams are involved. The discrete time MSTMM, however, has to use local linearization. In order to release the method from such approximations, a new version of MSTMM is presented in this paper where translational and angular accelerations, on the one hand, and internal forces and moments, on the other hand, are used as state variables. Already linear relationships among these quantities are utilized, which results in new element transfer matrices and algorithms making the study of multibody systems as simple as the study of single bodies. The proposed approach also allows combining MSTMM with any general numerical integration procedure. Some numerical examples of MSD are given to demonstrate the proposed method.     

Nonholonomic Multibody Dynamics

The paper deals with the nonholonomic multibody system dynamics from apoint of view which is caused by some actual applications in high-tecareas like high-speed train technology or biomechanics of somedisciplines in high-performance sports. Obviously, looking at suchproblems, there are very close connections between classical analyticaldynamics, differential geometry and modern control theory. But theseconnections cannot be used to get new composed results in solvingcomplicated problems of multibody system dynamics because correspondingsoftware tools are not enough in tune with each other. This paper willgive some ideas for developing a unified basis for modeling, simulationand control of nonholonomic multibody systems.First, a derivative-free approach for generating Lagrangian motionequations of multibody systems with kinematical tree structure as wellas for constrained multibody systems is given. This has been done byusing differential-geometric concepts in a Riemannian space. Secondly,the well-known theorem of Frobenius is considered with respect to itsclassical interpretation by the so-called object of nonholonomy as wellas by its modern interpretation in the nonlinear control theory usingLie-brackets. The ideas are illustrated by the classical rollingcondition and edge condition on double-curved surfaces. Specialnumerical problems in simulation of multibody systems subject toadditional kinematic constraints are discussed. Finally threeapplications are given.     

基于CORBA的多体系统动力学分布式仿真

基于多体系统动力学与运动学的机械系统计算机辅助分析(CAA)已成为该领域计算机辅助工程(CAE)的核心内容。该文分析了多体系统动力学方程建立及求解方法，结合CORBA技术、数值分析方法、OpeaGL图形技术，开发了基于CORBA的多体系统动力学分布式仿真系统，实现了分布式环境下的多体系统动力学的数值分析与仿真。     

Erratum to: New efficient method for dynamic modeling and simulation of flexible multibody systems moving in plane

Efficient, precise dynamic analysis for general flexible multibody systems has become a research focus in the field of flexible multibody dynamics. In this paper, the finite element method and component mode synthesis are introduced to describe the deformations of the flexible components, and the dynamic equations of flexible bodies moving in plane are deduced. By combining the discrete time transfer matrix method of multibody system with these dynamic equations of flexible component, the transfer equations and transfer matrices of flexible bodies moving in plane are developed. Finally, a high-efficient dynamic modeling method and its algorithm are presented for high-speed computation of general flexible multibody dynamics. Compared with the ordinary dynamics methods, the proposed method combines the strengths of the transfer matrix method and finite element method. It does not need the global dynamic equations of system and has the low order of system matrix and high computational efficiency. This method can be applied to solve the dynamics problems of flexible multibody systems containing irregularly shaped flexible components. It has advantages for dynamic design of complex flexible multibody systems. Formulations as well as a numerical example of a multi-rigid-flexible-body system containing irregularly shaped flexible components are given to validate the method.     

变拓扑柔性多体系统接触碰撞动力学研究

在实际工程领域中存在着大量接触碰撞等非连续动力学问题,现有的解决柔性多体系统连续动力学过程的建模理论与方法,已经无法解决或无法很好解决这些问题.本文基于变拓扑思想,提出了附加接触约束的柔性多体系统碰撞动力学建模理论；通过设计柔性圆柱杆接触碰撞实验,验证了所提出附加约束接触碰撞模型的有效性；针对柔性多体系统全局动力学仿真面临时间和空间的多尺度问题,提出多变量的离散方法,从而提高了柔性多体系统非连续动力学的仿真效率.     

New efficient method for dynamic modeling and simulation of flexible multibody systems moving in plane

Efficient, precise dynamic analysis for general flexible multibody systems has become a research focus in the field of flexible multibody dynamics. In this paper, the finite element method and component mode synthesis are introduced to describe the deformations of the flexible components, and the dynamic equations of flexible bodies moving in plane are deduced. By combining the discrete time transfer matrix method of multibody system with these dynamic equations of flexible component, the transfer equations and transfer matrices of flexible bodies moving in plane are developed. Finally, a high-efficient dynamic modeling method and its algorithm are presented for high-speed computation of general flexible multibody dynamics. Compared with the ordinary dynamics methods, the proposed method combines the strengths of the transfer matrix method and finite element method. It does not need the global dynamic equations of system and has the low order of system matrix and high computational efficiency. This method can be applied to solve the dynamics problems of flexible multibody systems containing irregularly shaped flexible components. It has advantages for dynamic design of complex flexible multibody systems. Formulations as well as a numerical example of a multi-rigid-flexible-body system containing irregularly shaped flexible components are given to validate the method.     

Transfer matrix method for linear multibody system

A new method for linear hybrid multibody system dynamics is proposed in this paper. This method, named as transfer matrix method of linear multibody system (MSTMM), expands the advantages of the traditional transfer matrix method (TMM). The concepts of augmented eigenvector and equation of motion of linear hybrid multibody system are presented at first to find the orthogonality and to analyze the responses of the hybrid multibody system using modal method. If using this method, the global dynamics equation is not needed in the study of linear hybrid multibody system dynamics. The MSTMM has a small size of matrix and higher computational speed, and can be applied to linear multi-rigid-body system dynamics, linear multi-flexible-body system dynamics and linear hybrid multibody system dynamics. This method is simple, straightforward, practical, and provides a powerful tool for the study on linear hybrid multibody system dynamics. This method can be used in the following: (1) Solve the eigenvalue problem of linear hybrid multibody systems. (2) Obtain the orthogonality of eigenvectors of linear hybrid multibody systems. (3) Realize the accurate analysis of the dynamics response of linear hybrid multibody systems. (4) Find the connected parameters between bodies used in the computation of linear hybrid multibody systems. A practical engineering system is taken as an example of linear multi-rigid-flexible-body system, the dynamics model, the transfer equations and transfer matrices of various bodies and hinges; the overall transfer equation and overall transfer matrix of the system are developed. Numerical example shows that the results of the vibration characteristics and the response of the hybrid multibody system received by MSTMM and by experiment have good agreements. These validate the proposed method.     

Performance of automatic differentiation tools in the dynamic simulation of multibody systems

Within the multibody systems literature, few attempts have been made to use automatic differentiation for solving forward multibody dynamics and evaluating its computational efficiency. The most relevant implementations are found in the sensitivity analysis field, but they rarely address automatic differentiation issues in depth. This paper presents a thorough analysis of automatic differentiation tools in the time integration of multibody systems. To that end, a penalty formulation is implemented. First, open-chain generalized positions and velocities are computed recursively, while using Cartesian coordinates to define local geometry. Second, the equations of motion are implicitly integrated by using the trapezoidal rule and a Newton–Raphson iteration. Third, velocity and acceleration projections are carried out to enforce kinematic constraints. For the computation of Newton–Raphson’s tangent matrix, instead of using numerical or analytical differentiation, automatic differentiation is implemented here. Specifically, the source-to-source transformation tool ADIC2 and the operator overloading tool ADOL-C are employed, in both dense and sparse modes. The theoretical approach is backed with the numerical analysis of a 1-DOF spatial four-bar mechanism, three different configurations of a 15-DOF multiple four-bar linkage, and a 16-DOF coach maneuver. Numerical and automatic differentiation are compared in terms of their computational efficiency and accuracy. Overall, we provide a global perspective of the efficiency of automatic differentiation in the field of multibody system dynamics.     

Deduction method of the overall transfer equation of linear controlled multibody systems

The transfer matrix method for multibody systems (MSTMM), which is a highly efficient and novel approach for multibody system dynamics, was proposed and perfected in the past 20 years. The deduction of the overall transfer equation of the system is one of the key techniques in MSTMM. The topology figure of the dynamics model of multibody systems is a novel pictorial expression to describe the relationship among the state vectors of connection points of different elements in MSTMM. In this paper, the block diagram in control theory is introduced and incorporated into the topology figure of the dynamics model to represent the connection relationship between different mechanical elements in the system as well as the control relations. Meanwhile, the transfer equations of the controlled element, control subsystem and the overall transfer equation of the linear controlled multibody systems are deduced. The proposed method greatly reduces the efforts to study the linear controlled multibody systems since the procedures are stylized. Two numerical examples are given to validate the proposed method.     

First order sensitivity analysis of flexible multibody systems using absolute nodal coordinate formulation

Design sensitivity analysis of flexible multibody systems is important in optimizing the performance of mechanical systems. The choice of coordinates to describe the motion of multibody systems has a great influence on the efficiency and accuracy of both the dynamic and sensitivity analysis. In the flexible multibody system dynamics, both the floating frame of reference formulation (FFRF) and absolute nodal coordinate formulation (ANCF) are frequently utilized to describe flexibility, however, only the former has been used in design sensitivity analysis. In this article, ANCF, which has been recently developed and focuses on modeling of beams and plates in large deformation problems, is extended into design sensitivity analysis of flexible multibody systems. The Motion equations of a constrained flexible multibody system are expressed as a set of index-3 differential algebraic equations (DAEs), in which the element elastic forces are defined using nonlinear strain-displacement relations. Both the direct differentiation method and adjoint variable method are performed to do sensitivity analysis and the related dynamic and sensitivity equations are integrated with HHT-I3 algorithm. In this paper, a new method to deduce system sensitivity equations is proposed. With this approach, the system sensitivity equations are constructed by assembling the element sensitivity equations with the help of invariant matrices, which results in the advantage that the complex symbolic differentiation of the dynamic equations is avoided when the flexible multibody system model is changed. Besides that, the dynamic and sensitivity equations formed with the proposed method can be efficiently integrated using HHT-I3 method, which makes the efficiency of the direct differentiation method comparable to that of the adjoint variable method when the number of design variables is not extremely large. All these improvements greatly enhance the application value of the direct differentiation method in the engineering optimization of the ANCF-based flexible multibody systems.     

A frequency domain approach for parameter identification in multibody dynamics

The adjoint method shows an efficient way to incorporate inverse dynamics to engineering multibody applications, as, e.g., parameter identification. In case of the identification of parameters in oscillating multibody systems, a combination of Fourier analysis and the adjoint method is an obvious and promising approach. The present paper shows the adjoint method including adjoint Fourier coefficients for the parameter identification of the amplitude response of oscillations. Two examples show the potential and efficiency of the proposed method in multibody dynamics.     

Flexible Multibody Dynamics: Review of Past and Recent Developments

In this paper, a review of past and recent developments in the dynamics of flexible multibody systems is presented. The objective is to review some of the basic approaches used in the computer aided kinematic and dynamic analysis of flexible mechanical systems, and to identify future directions in this research area. Among the formulations reviewed in this paper are the floating frame of reference formulation, the finite element incremental methods, large rotation vector formulations, the finite segment method, and the linear theory of elastodynamics. Linearization of the flexible multibody equations that results from the use of the incremental finite element formulations is discussed. Because of space limitations, it is impossible to list all the contributions made in this important area. The reader, however, can find more references by consulting the list of articles and books cited at the end of the paper. Furthermore, the numerical procedures used for solving the differential and algebraic equations of flexible multibody systems are not discussed in this paper since these procedures are similar to the techniques used in rigid body dynamics. More details about these numerical procedures as well as the roots and perspectives of multibody system dynamics are discussed in a companion review by Schiehlen [79]. Future research areas in flexible multibody dynamics are identified as establishing the relationship between different formulations, contact and impact dynamics, control-structure interaction, use of modal identification and experimental methods in flexible multibody simulations, application of flexible multibody techniques to computer graphics, numerical issues, and large deformation problem. Establishing the relationship between different flexible multibody formulations is an important issue since there is a need to clearly define the assumptions and approximations underlying each formulation. This will allow us to establish guidelines and criteria that define the limitations of each approach used in flexible multibody dynamics. This task can now be accomplished by using the absolute nodal coordinate formulation which was recently introduced for the large deformation analysis of flexible multibody systems.     

On the contact detection for contact-impact analysis in multibody systems

One of the most important and complex parts of the simulation of multibody systems with contact-impact involves the detection of the precise instant of impact. In general, the periods of contact are very small and, therefore, the selection of the time step for the integration of the time derivatives of the state variables plays a crucial role in the dynamics of multibody systems. The conservative approach is to use very small time steps throughout the analysis. However, this solution is not efficient from the computational view point. When variable time-step integration algorithms are used and the preimpact dynamics does not involve high-frequencies, the integration algorithms may use larger time steps and the contact between two surfaces may start with initial penetrations that are artificially high. This fact leads either to a stall of the integration algorithm or to contact forces that are physically impossible which, in turn, lead to post-impact dynamics that is unrelated to the physical problem. The main purpose of this work is to present a general and comprehensive approach to automatically adjust the time step, in variable time-step integration algorithms, in the vicinity of contact of multibody systems. The proposed methodology ensures that for any impact in a multibody system the time step of the integration is such that any initial penetration is below any prescribed threshold. In the case of the start of contact, and after a time step is complete, the numerical error control of the selected integration algorithm is forced to handle the physical criteria to accept/reject time steps in equal terms with the numerical error control that it normally uses. The main features of this approach are the simplicity of its computational implementation, its good computational efficiency, and its ability to deal with the transitions between non-contact and contact situations in multibody dynamics. A demonstration case provides the results that support the discussion and show the validity of the proposed methodology.     

A complete strategy for efficient and accurate multibody dynamics of flexible structures with large lap joints considering contact and friction

This paper deals with the dynamics of jointed flexible structures in multibody simulations. Joints are areas where the surfaces of substructures come into contact, for example, screwed or bolted joints. Depending on the spatial distribution of the joint, the overall dynamic behavior can be influenced significantly. Therefore, it is essential to consider the nonlinear contact and friction phenomena over the entire joint. In multibody dynamics, flexible bodies are often treated by the use of reduction methods, such as component mode synthesis (CMS). For jointed flexible structures, it is important to accurately compute the local deformations inside the joint in order to get a realistic representation of the nonlinear contact and friction forces. CMS alone is not suitable for the capture of these local nonlinearities and therefore is extended in this paper with problem-oriented trial vectors. The computation of these trial vectors is based on trial vector derivatives of the CMS reduction base. This paper describes the application of this extended reduction method to general multibody systems, under consideration of the contact and friction forces in the vector of generalized forces and the Jacobian. To ensure accuracy and numerical efficiency, different contact and friction models are investigated and evaluated. The complete strategy is applied to a multibody system containing a multilayered flexible structure. The numerical results confirm that the method leads to accurate results with low computational effort.     

Multibody system dynamics simulator for process simulation of ships and offshore plants in shipyards

Since various existing simulation tools based on multibody system dynamics focus on conventional mechanical systems, such as machinery, cars, and spacecraft, there are some problems with the application of such simulation tools to shipbuilding domains due to the absence of specific items in the field of naval architecture and ocean engineering, such as hydrostatics, hydrodynamics, and mooring forces. Thus, in this study, we developed a multibody system dynamics simulator for the process simulation of ships and offshore structures. We based the simulator on six kernels: the multibody system dynamics kernel, the force calculation kernel, the numerical analysis kernel, the hybrid simulation kernel, the scenario management kernel, and the collision detection kernel. Based on these kernels, we implemented a simulator that had the following Graphic User Interfaces (GUIs): the modeling, visualization, and report GUIs. In addition, the geometric properties of blocks and facilities in shipyards are needed to configure the simulation for the production of ships and offshore plants, so these are managed in a database and connected to a specific commercial CAD system in shipyards. We used the simulator we developed in various cases of the process simulation of ships and offshore plants. The results show that the simulator is useful for various simulations of operations in shipyards and offshore industries.     

多柔体系统动力学研究进展与挑战

首先回顾多体系统动力学的学科发展和学术交流情况,然后系统概述了多柔体系统动力学方程数值算法、多柔体系统接触/碰撞动力学与柔性空间结构展开动力学三个方面的研究进展及值得关注的若干问题,最后给出了开展多柔体系统动力学研究的若干建议.     

Discrete Time Transfer Matrix Method for Multibody System Dynamics

A new method for multibody system dynamics is proposed in this paper. This method, named as discrete time transfer matrix method of multibody system (MS-DT-TMM), combines and expands the advantages of the transfer matrix method (TMM), transfer matrix method of vibration of multibody system (MS-TMM), discrete time transfer matrix method (DT-TMM) and the numerical integration procedure. It does not need the global dynamics equations for the study of multibody system dynamics. It has the modeling flexibility and a small size of matrices, and can be applied to a wide range of problems including multi-rigid-body system dynamics and multi-flexible-body system dynamics. This method is simple, straightforward, practical, and provides a powerful tool for the study of multibody system dynamics. Formulations of the method as well as some numerical examples of multi-rigid-body system dynamics and multi-flexible-body system dynamics to validate the method are given.     

Absolute nodal coordinate plane beam formulation for multibody systems dynamics

A new plane beam dynamic formulation for constrained multibody system dynamics is developed. Flexible multibody system dynamics includes rigid body dynamics and superimposed vibratory motions. The complexity of mechanical system dynamics originates from rotational kinematics, but the natural coordinate formulation does not use rotational coordinates, so that simple dynamic formulation is possible. These methods use only translational coordinates and simple algebraic constraints. A new formulation for plane flexible multibody systems are developed utilizing the curvature of a beam and point masses. Using absolute nodal coordinates, a constant mass matrix is obtained and the elastic force becomes a nonlinear function of the nodal coordinates. In this formulation, no infinitesimal or finite rotation assumptions are used and no assumption on the magnitude of the element rotations is made. The distributed body mass and applied forces are lumped to the point masses. Closed loop mechanical systems consisting of elastic beams can be modeled without constraints since the loop closure constraints can be substituted as beam longitudinal elasticity. A curved beam is modeled automatically. Several numerical examples are presented to show the effectiveness of this method.     

Non-linear dynamics of flexible multibody systems

In this work we set to examine several important issues pertinent to currently very active research area of the finite element modeling of flexible multibody system dynamics. To that end, we first briefly introduce three different model problems in non-linear dynamics of flexible 3D solid, a rigid body and 3D geometrically exact beam, which covers the vast majority of representative models for the particular components of a multibody system. The finite element semi-discretization for these models is presented along with the time-discretization performed by the mid-point scheme. In extending the proposed methodology to modeling of flexible multibody systems, we also present how to build a systematic representation of any kind of joint connecting two multibody components, a typical case of holonomic contraint, as a linear superposition of elementary constraints. We also indicate by a chosen model of rolling contact, an example of non-holonomic constraint, that the latter can also be included within the proposed framework. An important aspect regarding the reduction of computational cost while retaining the consistency of the model is also addressed in terms of systematic use of the rigid component hypothesis, mass lumping and the appropriate application of the explicit-implicit time-integration scheme to the problem on hand. Several numerical simulations dealing with non-linear dynamics of flexible multibody systems undergoing large overall motion are presented to further illustrate the potential of presented methodology. Closing remarks are given to summarize the recent achievements and point out several directions for future research.